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Metamath Proof Explorer

Theorem erngmul

Description: Ring addition operation. (Contributed by NM, 10-Jun-2013)

Ref Expression
Hypotheses erngset.h 
|- H = ( LHyp ` K )
erngset.t 
|- T = ( ( LTrn ` K ) ` W )
erngset.e 
|- E = ( ( TEndo ` K ) ` W )
erngset.d 
|- D = ( ( EDRing ` K ) ` W )
erng.m 
|- .x. = ( .r ` D )
Assertion erngmul 
|- ( ( ( K e. X /\ W e. H ) /\ ( U e. E /\ V e. E ) ) -> ( U .x. V ) = ( U o. V ) )

Proof

Step Hyp Ref Expression
1 erngset.h 
 |-  H = ( LHyp ` K )
2 erngset.t 
 |-  T = ( ( LTrn ` K ) ` W )
3 erngset.e 
 |-  E = ( ( TEndo ` K ) ` W )
4 erngset.d 
 |-  D = ( ( EDRing ` K ) ` W )
5 erng.m 
 |-  .x. = ( .r ` D )
6 1 2 3 4 5 erngfmul 
 |-  ( ( K e. X /\ W e. H ) -> .x. = ( s e. E , t e. E |-> ( s o. t ) ) )
7 6 oveqd 
 |-  ( ( K e. X /\ W e. H ) -> ( U .x. V ) = ( U ( s e. E , t e. E |-> ( s o. t ) ) V ) )
8 coexg 
 |-  ( ( U e. E /\ V e. E ) -> ( U o. V ) e. _V )
9 coeq1 
 |-  ( s = U -> ( s o. t ) = ( U o. t ) )
10 coeq2 
 |-  ( t = V -> ( U o. t ) = ( U o. V ) )
11 eqid 
 |-  ( s e. E , t e. E |-> ( s o. t ) ) = ( s e. E , t e. E |-> ( s o. t ) )
12 9 10 11 ovmpog 
 |-  ( ( U e. E /\ V e. E /\ ( U o. V ) e. _V ) -> ( U ( s e. E , t e. E |-> ( s o. t ) ) V ) = ( U o. V ) )
13 8 12 mpd3an3 
 |-  ( ( U e. E /\ V e. E ) -> ( U ( s e. E , t e. E |-> ( s o. t ) ) V ) = ( U o. V ) )
14 7 13 sylan9eq 
 |-  ( ( ( K e. X /\ W e. H ) /\ ( U e. E /\ V e. E ) ) -> ( U .x. V ) = ( U o. V ) )